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Dirac Operator

TIFRMumbai, India

Paul BaumPenn State

3 August, 2015

ATIYAH-SINGER INDEX THEOREM

There are several proofs of the Atiyah-Singer index theorem :

The first proof by M. F. Atiyah and I. M. Singer.Published in the R. Palais book Seminar on the Atiyah-SingerIndex Theorem, Princeton University Press (1965).The proof uses cobordism theory i.e. Pontrjagin-Thomconstruction and calculation of cobordism groups.A number of changes-simplifications in the proof have beenmade by M. S. Raghunathan.M. S. RaghunathanThe Atiyah-Singer Index Theorem,published in “Contemporary Mathematics”, Vol 522, (VectorBundles and Complex Geometry) (2008).

The K-theory proof by Atiyah-Segal-Singer.Five papers in Ann. of Math. (1968) (1971).Proves Atiyah-Singer for the equivariant caseand the families case.

The heat equation proof by Atiyah-Bott-Patodi.On the heat equation and the index theorem,Invent. Math. 19 (4): 279 - 330, (1973).Based on an idea originally proposed by McKean and Singer.Uses Chern-Weil theory.Does not in and of itself prove the general case ofAtiyah-Singer — but proves a very strong result for Diracoperators. Sets the stage for the eta-invariant and indextheory on compact manifolds with boundary.

Bott periodicity proof by Baum - van Erp.Reveals exactly how and why the Atiyah-Singer index theoremcan be proved as a corollary of Bott periodicity.Does not use Pontrjagin-Thom construction and calculationof cobordism groups.

Tangent groupoid proof by Alain Connes.Published in the book Noncommutative Geometry by AlainConnes, Academic Press, (1994).See also the cyclic cohomology theory of Alain Connes.

Bivariant K-theory proof by Joachim Cuntz.Uses Kasparov bivariant K-theory.

Five lectures:

1. Dirac operator

2. Atiyah-Singer revisited

3. What is K-homology?

4. Beyond ellipticity

5. The Riemann-Roch theorem

DIRAC OPERATOR

The Dirac operator of Rn will be defined. This is a first orderelliptic differential operator with constant coefficients. Next, theclass of differentiable manifolds which come equipped with anorder one differential operator which at the symbol level is locallyisomorphic to the Dirac operator of Rn will be considered. Theseare the Spinc manifolds. Spinc is slightly stronger than oriented, soSpinc can be viewed as “oriented plus epsilon”. Most of theoriented manifolds that occur in practice are Spinc. The Diracoperator of a closed Spinc manifold is the basic example for theHirzebruch-Riemann-Roch theorem and the Atiyah-Singer indextheorem.

What is the Dirac operator of Rn?

To answer this, shall construct matrices E1, E2, . . . , En with thefollowing properties :

Properties of E1, E2, . . . , En

Each Ej is a 2r × 2r matrix of complex numbers,where r is the largest integer ≤ n/2.

Each Ej is skew adjoint, i.e. E∗j = −Ej(* = conjugate transpose)

E2j = −I j = 1, 2, . . . , n

(I is the 2r × 2r identity matrix.)

EjEk + EkEj = 0 whenever j 6= k.

For n odd, (n = 2r + 1) ir+1E1E2 · · ·En = I i =√−1

For n even, (n = 2r) each Ej is of the form

Ej = [ 0 ∗∗ 0 ] and irE1E2 · · ·En =[I 00 −I

]

These matrices are constructed by a simple inductive procedure.

n = 1, E1 = [−i]———————————————————————————n n+ 1 with n odd (r r + 1)

The new matrices E1, E2, . . . , En+1 are

Ej =[

0 EjEj 0

]for j = 1, . . . , n and En+1 =

[0 −II 0

]where E1, E2, . . . , En are the old matrices.———————————————————————————–n n+ 1 with n even (r does not change)

The new matrices E1, E2, . . . , En+1 are

Ej = Ej for j = 1, . . . , n and En+1 =[−iI 0

0 iI

]where E1, E2, . . . , En are the old matrices.

Example

n = 1: E1 = [−i]

n = 2: E1 =[

0 −i−i 0

], E2 =

[0 −11 0

]n = 3: E1 =

[0 −i−i 0

], E2 =

[0 −11 0

], E3 =

[−i 00 i

]

Example

n = 4: E1 =

[0 0 0 −i0 0 −i 00 −i 0 0−i 0 0 0

]E2 =

[0 0 0 −10 0 1 00 −1 0 01 0 0 0

]

E3 =

[0 0 −i 00 0 0 i−i 0 0 00 i 0 0

]E4 =

[0 0 −1 00 0 0 −11 0 0 00 1 0 0

]

D = Dirac operator of Rnn = 2r n evenn = 2r + 1 n odd

D =

n∑j=1

Ej∂

∂xj

D is an unbounded symmetric operator on the Hilbert spaceL2(Rn)⊕ L2(Rn)⊕ . . .⊕ L2(Rn) (2r times)

To begin, the domain of D isC∞c (Rn)⊕ C∞c (Rn)⊕ . . .⊕ C∞c (Rn) (2r times)

D is essentially self-adjoint(i.e. D has a unique self-adjoint extension)so it is natural to view D as an unbounded self-adjoint operatoron the Hilbert spaceL2(Rn)⊕ L2(Rn)⊕ . . .⊕ L2(Rn) (2r times)

QUESTION : Let M be a C∞ manifold of dimension n.Does M admit a differential operator which (at the symbol level)is locally isomorphic to the Dirac operator of Rn?

To answer this question, will define Spinc vector bundle.

What is a Spinc vector bundle?

Let X be a paracompact Hausdorff topological space.On X let E be an R vector bundle which has been oriented.i.e. the structure group of E has been reduced fromGL(n,R) to GL+(n,R)

GL+(n,R) = [aij ] ∈ GL(n,R) | det[aij ] > 0

n= fiber dimension (E)

Assume n ≥ 3 and recall that for n ≥ 3

H2(GL+(n,R);Z) = Z/2Z

Denote by F+(E) the principal GL+(n,R) bundle on X consistingof all positively oriented frames.

A point of F+(E) is a pair(x, (v1, v2, . . . , vn)

)where x ∈ X

and (v1, v2, . . . , vn) is a positively oriented basis of Ex. Theprojection F+(E)→ X is(

x, (v1, v2, . . . , vn))7→ x

For x ∈ X, denote by

ιx : F+x (E) → F+(E)

the inclusion of the fiber at x into F+(E).

Note that (with n ≥ 3)

H2(F+x (E);Z) = Z/2Z

A Spinc vector bundle on X is an R vector bundle E on X(fiber dimension E ≥ 3) with

1 E has been oriented.

2 α ∈ H2(F+(E);Z) has been chosen such that ∀x ∈ X

ι∗x(α) ∈ H2(F+x (E);Z) is non-zero.

Remarks

1.For n = 1, 2 “E is a Spinc vector bundle” =“E has been orientedand an element α ∈ H2(X;Z)” has been chosen. (α can be zero.)

2. For all values of n = fiber dimension(E), E is a Spinc vectorbundle iff the structure group of E has been changed fromGL(n,R) to Spinc(n).i.e. Such a change of structure group is equivalent to the abovedefinition of Spinc vector bundle.

By forgetting some structure a complex vector bundle or a Spinvector bundle canonically becomes a Spinc vector bundle

complex⇓

Spin ⇒ Spinc

⇓oriented

A Spinc structure for an R vector bundle E can be thought of asan orientation for E plus a slight extra bit of structure. Spinc

structures behave very much like orientations. For example, anorientation on two out of three R vector bundles in a short exactsequence determines an orientation on the third vector bundle. Ananalogous assertion is true for Spinc structures.

Two Out Of Three Lemma

Lemma

Let0 −→ E′ −→ E −→ E′′ −→ 0

be a short exact sequence of R-vector bundles on X. If two out ofthree are Spinc vector bundles, then so is the third.

Definition

Let M be a C∞ manifold (with or without boundary). M is aSpinc manifold iff the tangent bundle TM of M is a Spinc vectorbundle on M .

The Two Out Of Three Lemma implies that the boundary ∂M ofa Spinc manifold M with boundary is again a Spinc manifold.

Various well-known structures on a manifold M make M into aSpinc manifold.

(complex-analytic)⇓

(symplectic) ⇒ (almost complex)⇓

(contact) ⇒ (stably almost complex)⇓

Spin ⇒ Spinc

⇓(oriented)

A Spinc manifold can be thought of as an oriented manifold with aslight extra bit of structure. Most of the oriented manifolds whichoccur in practice are Spinc manifolds.

A Spinc manifold comes equipped with a first-order ellipticdifferential operator known as its Dirac operator. This operator islocally isomorphic (at the symbol level) to the Dirac operator ofRn.

EXAMPLE. Let M be a compact complex-analytic manifold.Set Ωp,q = C∞(M,Λp,qT ∗CM)Ωp,q is the C vector space of all C∞ differential forms of type (p, q)Dolbeault complex

0 −→ Ω0,0 −→ Ω0,1 −→ Ω0,2 −→ · · · −→ Ω0,n −→ 0

The Dirac operator (of the underlying Spinc manifold) is theassembled Dolbeault complex

∂ + ∂∗ :⊕j

Ω0, 2j −→⊕j

Ω0, 2j+1

The index of this operator is the arithmetic genus of M — i.e. isthe Euler number of the Dolbeault complex.

TWO POINTS OF VIEW ON SPINc MANIFOLDS

1. Spinc is a slight strengthening of oriented. Most of the orientedmanifolds that occur in practice are Spinc.

2. Spinc is much weaker than complex-analytic. BUT theassempled Dolbeault complex survives (as the Dirac operator).AND the Todd class survives.

M Spinc =⇒ ∃ Td(M) ∈ H∗(M ;Q)

If M is a Spinc manifold, then Td(M) is

Td(M) := expc1(M)/2A(M) Td(M) ∈ H∗(M ;Q)

If M is a complex-analyic manifold, then M has Chern classesc1, c2, . . . , cn and

expc1(M)/2A(M) = PTodd(c1, c2, . . . , cn)

WARNING!!!

The Todd class of a Spinc manifold is not obtained bycomplexifying the tangent bundle TM of M and thenapplying the Todd polynomial to the Chern classes of TCM .

Td(TCM) = A(M)2 = A(M) ∪ A(M)

Correct formula for the Todd class of a Spinc manifold M is:


SPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .

DE denotes the Dirac operator of M tensored with E.

DE : C∞(M,S+ ⊗ E) −→ C∞(M,S− ⊗ E)

S+, (S−) are the positive (negative) spinor bundles on M .

THEOREM Index(DE) = (ch(E) ∪ Td(M))[M ].

SPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .DE denotes the Dirac operator of M tensored with E.


This theorem will be proved in the next lecture as a corollary ofBott periodicity.

In particular, this will prove the Hirzebruch-Riemann-Rochtheorem.

Also, this will prove (for closed even-dimensional Spinc manifolds)the Hirzebruch signature theorem.

From E1, E2, . . . , En obtain :

1) The Dirac operator of Rn

2) The Bott generator vector bundle on Sn (n even)

3) The spin representation of Spinc(n)

W finite dimensional C vector space dimC(W ) <∞

T : W →W T ∈ HomC(W,W ) T 2 = I

=⇒ The eigen-values of T are ± 1

W = W1 ⊕W−1

W1 = v ∈W | Tv = v W−1 = v ∈W | Tv = −v

Bott generator vector bundle

n even n = 2r Sn ⊂ Rn+1 Sn →M(2r,C)

Sn = (a1, a2, . . . , an+1) ∈ Rn+1 | a21 + a2

2 + · · ·+ a2n+1 = 1

(a1, a2, . . . , an+1) 7→ i(a1E1 + a2E2 + · · ·+ an+1En+1)i =√−1

(i)2(a1E1 + a2E2 + · · ·+ an+1En+1)2

= (−1)(−a21 − a2

2 − . . .− a2n+1) I

= I

=⇒ The eigenvalues of i(a1E1 + a2E2 + · · ·+ an+1En+1) are ± 1.

Bott generator vector bundle β on Sn n even n = 2r

β(a1,a2,...,an+1)

= (+1 eigenspace of i(a1E1 + a2E2 + · · ·+ an+1En+1))∗

= HomC(v ∈ C2r | i(a1E1 + a2E2 + · · ·+ an+1En+1) v = v , C)

K0(Sn) = Z ⊕ Z1 β

1 = Sn × C

Bott generator vector bundle β on Sn n even n = 2rβ is determined by:

1 ∀p ∈ Sn , dimC(βp) = 2r−1

2 ch(β)[Sn] = 1

n even n = 2r Sn ⊂ Rn+1

With the Spin (or Spinc) structure Sn has as the boundary of theunit ball Bn+1 of Rn+1, the Spinor bundle S of Sn is:

S = Sn × C2r

The positive (negative) Spinor bundles S+(S−) are defined by :

S+(a1,a2,...,an+1) = +1 eigenspace of i(a1E1+a2E2+· · ·+an+1En+1)

S−(a1,a2,...,an+1) = −1 eigenspace of i(a1E1+a2E2+· · ·+an+1En+1)

S = Sn × C2r = S+ ⊕ S−

β = (S+)∗

M Spinc manifold

∂M might be non-empty

TM = the tangent bundle of M

Dirac operatorD : C∞c (M,S)→ C∞c (M,S)

S is the Spinor bundle

C∞c (M,S) = C∞ sections with compact support of S

D : C∞c (M,S)→ C∞c (M,S)

such that

(1) D is C-linearD(s1 + s2) = Ds1 +Ds2 sj ∈ C∞c (M,S)D(λx) = λDs λ ∈ C

(2) If f : M → C is a C∞ function, then

D(fs) = (df)s+ f(Ds)

(3) If sj ∈ C∞c (M,S) then∫M

(Ds1x, s2x) =

∫M

(s1x,Ds2x)dx

(4) If dim M is even, then D is off-diagonal S = S+ ⊕ S−

D =

0 D−

D+ 0

D : C∞c (M,S)→ C∞c (M,S) is an elliptic first-order differentialoperator.

D can be viewed as an unbounded operator on the Hilbert spaceL2(M,S)

(s1, s2) =

∫M

(s1x, s2x)dx

D : C∞c (M,S)→ C∞c (M,S)

is a symmetric operator

EXAMPLE. Let M be a compact complex-analytic manifold.The positive (negative) Spinor bundles of the underlying Spinc

manifold are :S+ =

⊕j

Λ0, 2j T ∗CM

S− =⊕j

Λ0, 2j+1 T ∗CM

D+ : C∞(M,S+)→ C∞(M,S−) is

∂ + ∂∗ : C∞(M,⊕j

Λ0, 2j T ∗CM) −→ C∞(M,⊕j

Λ0, 2j+1 T ∗CM)


EXAMPLE. Let M be a compact even-dimensional Spinc manifoldwithout boundary.

D+S∗ : C∞(M,S+ ⊗ S∗) −→ C∞(M,S− ⊗ S∗)

is the Hirzebruch signature operator of M .If the dimension of M is divisible by 4, the index of this operator isthe signature of the quadratic form

Hr(M ;R)⊗R Hr(M,R) −→ R n = 2r r even

a⊗ b 7→ (a ∪ b)[M ]

Example.n even n = 2r Sn ⊂ Rn+1

D = Dirac operator of Sn

S = Spinor bundle of Sn = Sn × C2r

S = S+ ⊕ S−

D : C∞(Sn,S)→ C∞(Sn,S)

D =

0 D−

D+ 0

D+ : C∞(Sn,S+)→ C∞(Sn,S−)

Index (D+) := dimC(KernelD+)− dimC(CokernelD+) = 0

Theorem. Index (D+) = 0

Tensor D+ with the Bott generator vector bundle β

D+β : C∞(Sn,S+ ⊗ β)→ C∞(Sn,S− ⊗ β)

Theorem. On Sn, with n even,Index(D+) = 0 and Index (D+

β ) = 1.

BOTT PERIODICITY

πjGL(n,C) =

Z j odd

0 j even

j = 0, 1, 2, . . . , 2n− 1

Why ???? does Bott periodicity implySPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .DE denotes the Dirac operator of M tensored with E.


This will be explained in the next lecture — tomorrow(i.e. Tuesday, 4 August).

Two out of three lemma.

Let 0→ E′ → E → E′′ → 0 be an exact sequence of R vectorbundles on X. If Spinc structures are given for any two of E′, E,E′′ then a Spinc structure is determined for the third.

Corollary. If M is a Spinc manifold with boundary ∂M , then ∂Mis (in a canonical way) a Spinc manifold.

Proof.

set 1 = ∂M × Rexact sequence

0→ T (∂M)→ TM | ∂M → 1→ 0

By forgetting some structure a complex vector bundle or a Spinvector bundle canonically becomes a Spinc vector bundle

complex⇓

Spin ⇒ Spinc

⇓oriented

A Spinc structure for the R vector bundle E can be thought of asan orientation for E plus a slight extra bit of structure. Spinc

structures behave very much like orientations. For example, anorientation on two out of three R vector bundles in a short exactsequence determines an orientation on the third vector bundle.Analogous assertions are true for Spinc structures.

Two out of three lemma.

Let 0→ E′ → E → E′′ → 0 be an exact sequence of R vectorbundles on X. If Spinc structures are given for any two of E′, E,E′′ then a Spinc structure is determined for the third.

Various well-known structures on a manifold M make M into aSpinc manifold




Spin ⇒ Spinc

⇓(oriented)


CliffC(V ) = C⊗R

Cliff(V )

V ⊂ Cliff(V ) ⊂ CliffC(V )

CliffC(V ) is a C∗ algebra

v ∈ V v∗ = −v

Choose an orthogonal basis e1, e2, . . . , en for V

n = dimR(V )

n even n = 2r

E1, E2, . . . , En 2r × 2r matrices

ej 7→ Ej CliffC(V ) −→∼= M(2r,C)

Isomorphism of C∗ algebras

n odd n = 2r + 1 n = dimR(V )

ϕ+ : CliffC(V )→M(2r,C)

E1, E2, . . . , En 2r × 2r matrices

ϕ+(ej) = Ej j = 1, 2, . . . , n

ϕ− : CliffC(V )→M(2r,C)

ϕ−(ej) = −Ej j = 1, 2, . . . , n

ϕ+ ⊕ ϕ− : CliffC(V ) −→∼= M(2r,C)⊕M(2r,C)

Isomorphism of C∗ algebras

Remark. These isomorphisms are non-canonical since they dependon the choice of an orthornormal basis for V .

Assume

X ×G→ XG acts on X by a right action

G× Y → YG acts on Y by a left action

Notation. X ×GY = X × Y/ ∼ (xg, y) ∼ (x, gy)

Example. E R vector bundle on X

∆(E) ×GL(n,R)

Rn ∼= E

((p, v1, v2, . . . , vn), (a1, a2, . . . , an)) 7→ a1v1 + a2v2 + · · ·+ anvn

E R vector bundle on X

A Spinc datum η : P → ∆(E) determinesa Spinor system (ε, 〈 , 〉, F ) for E.

ε and 〈 , 〉 p ∈ X An R basis v1, v2, . . . , vn of Ep is positivelyoriented and orthonormal iff

(v1, v2, . . . , vn) ∈ Image (η)

Spinor bundle F

n = 2r or n = 2r + 1

F = P ×Spinc(n)

C2r

How does Spinc(n) act on C2r ?

n odd

Spinc(n) has an irreducible representation known as its spinrepresentation

Spinc(n)→ GL(2r,C)

n = 2r + 1

n even

Spinc(n) has two irreducible representations known as its 1/2-spinrepresentations

Spinc(n)→ GL(2r−1,C)

Spinc(n)→ GL(2r−1,C)

The direct sum Spinc(n)→ GL(2r,C) of these tworepresentations is the spin representation of Spinc(n)

Consider Rn with its usual inner product and usual orthonormalbasis e1, e2, . . . , en

ϕ : CliffC(Rn)→M(2r,C)

ϕ(ej) = Ej j = 1, 2, . . . , n

There is a canonical inclusion

Spinc(n) ⊂ CliffC(Rn)

ϕ : CliffC(Rn)→M(2r,C) restricted to Spinc(n) maps Spinc(n)to 2r × 2r unitary matrices

Spinc(n)→ U(2r) ⊂ GL(2r,C)

This is the Spin representation of Spinc(n)

Spinc(n) acts on C2r via this representation

M C∞ manifold

∂M might be non-empty

TM = the tangent bundle of M

(Spinc datum for TMη : P → ∆(TM)

)y

(Spinor system for TM

(ε, 〈 , 〉, F )

)y

(Dirac operator

D : C∞c (M,F )→ C∞c (M,F )

)

F is the Spinor bundle

C∞c (M,F ) = C∞ sections with compact support of F

D : C∞c (M,F )→ C∞c (M,F )

such that

(1) D is C-linearD(s1 + s2) = Ds1 +Ds2 sj ∈ C∞c (M,F )D(λx) = λDs λ ∈ C

(2) If f : M → C is a C∞ function, then

D(fs) = (df)s+ f(Ds)

(3) If sj ∈ C∞c (M,F ) then∫M

(Ds1x, s2x) =

∫M

(s1x,Ds2x)dx

(4) If dim M is even, then D is off-diagonal F = F+ ⊕ F−

D =

0 D−

D+ 0

Existence of D ?

YES – Construct D locally and patch together with a C∞ partitionof unity.

Uniqueness of D ?

YES – If D0 and D1 both satisfy (1)-(4) then D0 −D1 is a vectorbundle map

D0 −D1 : F → F

Hence D0 and D1 differ by lower order terms

D : C∞c (M,F )→ C∞c (M,F ) is an elliptic first-order differentialoperator.

D can be viewed as an unbounded operator on the Hilbert spaceL2(M,F )

(s1, s2) =

∫M

(s1x, s2x)dx

D : C∞c (M,F )→ C∞c (M,F )

is a symmetric operator

Example.n even

Sn ⊂ Rn+1

D = Dirac operator of Sn

F = Spinor bundle of Sn

F = F+ ⊕ F−

D : C∞(Sn, F )→ C∞(Sn, F )

D =

0 D−

D+ 0

D+ : C∞(Sn, F+)→ C∞(Sn, F−)

Index (D+) := dimC(KernelD+)− dimC(CokernelD+)

Theorem. Index (D+) = 0

Tensor D+ with the Bott generator vector bundle β

D+β : C∞(Sn, F+ ⊗ β)→ C∞(Sn, F− ⊗ β)

Theorem. Index (D+β ) = 1

Atiyah-Singer Index theorem

M compact C∞ manifold without boundary

D an elliptic differential (or pseudo-differential) operator on M

E0, E1, C∞ C vector bundles on M

C∞(M,Ej) denotes the C vector space of all C∞ sections of Ej .

D : C∞(M,E0) −→ C∞(M,E1)

D is a linear transformation of C vector spaces.

Atiyah-Singer Index theorem

M compact C∞ manifold without boundary

D an elliptic differential (or pseudo-differential) operator on M

Index(D) := dimC (Kernel D)− dimC (Cokernel D)

Theorem (M.Atiyah and I.Singer)

Index (D) = (a topological formula)

Example

M = S1 = (t1, t2) ∈ R2 | t21 + t22 = 1

Df : L2(S1) −→ L2(S1) is

0

Tf

I

0

where L2(S1) = L2+(S1)⊕ L2

−(S1).

L2+(S1) has as orthonormal basis einθ with n = 0, 1, 2, . . .

L2−(S1) has as orthonormal basis einθ with n = −1,−2,−3, . . ..

Example

f : S1 −→ R2 − 0 is a C∞ map.

S1 R2 − (0, 0)f

Tf : L2+(S1) −→ L2

+(S1) is the composition

L2+(S1)

Mf−→ L2(S1) −→ L2+(S1)

Tf : L2+(S1) −→ L2

+(S1) is the Toeplitz operator associated to f

Example

Thus Tf is composition

Tf : L2+(S1)

Mf−→ L2(S1)P−→ L2

+(S1)

where L2+(S1)

Mf−→ L2(S1) is v 7→ fvfv(t1, t2) := f(t1, t2)v(t1, t2) ∀(t1, t2) ∈ S1 R2 = C

and L2(S1)P−→ L2

+(S1) is the Hilbert space projection.

Df (v + w) := Tf (v) + w v ∈ L2+(S1), w ∈ L2

−(S1)

Index(Df ) = -winding number (f).

RIEMANN - ROCH

M compact connected Riemann surface

genus of M = # of holes

=1

2[rankH1(M ;Z)]

D a divisor of M

D consists of a finite set of points of M p1, p2, . . . , pl and aninteger assigned to each point n1, n2, . . . , nl

Equivalently

D is a function D : M → Z with finite support

Support(D) = p ∈M | D(p) 6= 0

Support(D) is a finite subset of M

D a divisor on M

deg(D) :=∑

p∈M D(p)

Remark

D1, D2 two divisors

D1 = D2 iff ∀p ∈M,D1(p) = D2(p)

Remark

D a divisor, −D is(−D)(p) = −D(p)

Example

Let f : M → C ∪ ∞ be a meromorphic function.

Define a divisor δ(f) by:

δ(f)(p) =

0 if p is neither a zero nor a pole of f

order of the zero if f(p) = 0

−(order of the pole) if p is a pole of f

Example

Let w be a meromorphic 1-form on M . Locally w is f(z)dz wheref is a (locally defined) meromorphic function. Define a divisorδ(w) by:

δ(w)(p) =

0 if p is neither a zero nor a pole of w

order of the zero if w(p) = 0

−(order of the pole) if p is a pole of w

D a divisor on M

H0(M,D) :=

meromorphic functions

f : M → C ∪ ∞

∣∣∣∣∣ δ(f) = −D

H1(M,D) :=

meromorphic 1-forms

w on M

∣∣∣∣∣ δ(w) = D

Lemma

H0(M,D) and H1(M,D) are finite dimensional C vector spaces

dimCH0(M,D) <∞

dimCH1(M,D) <∞

Theorem (R. R.)

Let M be a compact connected Riemann surface and let D be adivisor on M . Then:

dimCH0(M,D)− dimCH

1(M,D) = d− g + 1

d = degree (D)

g = genus (M)

HIRZEBRUCH-RIEMANN-ROCH

M non-singular projective algebraic variety / CE an algebraic vector bundle on M

E = sheaf of germs of algebraic sections of E

Hj(M,E) := j-th cohomology of M using E,j = 0, 1, 2, 3, . . .

LEMMAFor all j = 0, 1, 2, . . . dimCH

j(M,E) <∞.

For all j > dimC(M), Hj(M,E) = 0.

χ(M,E) :=

n∑j=0

(−1)j dimCHj(M,E)

n = dimC(M)

THEOREM[HRR] Let M be a non-singular projective algebraicvariety / C and let E be an algebraic vector bundle on M . Then

χ(M,E) = (ch(E) ∪ Td(M))[M ]

Hirzebruch-Riemann-Roch

Theorem (HRR)

Let M be a non-singular projective algebraic variety / C and let Ebe an algebraic vector bundle on M . Then

χ(M,E) = (ch(E) ∪ Td(M))[M ]

Various well-known structures on a C∞ manifold M make M intoa Spinc manifold




Spin ⇒ Spinc

⇓(oriented)


Two Out Of Three Lemma

Lemma

Let0 −→ E′ −→ E −→ E′′ −→ 0

be a short exact sequence of R-vector bundles on X. If two out ofthree are Spinc vector bundles, then so is the third.

Definition

Let M be a C∞ manifold (with or without boundary). M is aSpinc manifold iff the tangent bundle TM of M is a Spinc vectorbundle on M .

The Two Out Of Three Lemma implies that the boundary ∂M ofa Spinc manifold M with boundary is again a Spinc manifold.

A Spinc manifold comes equipped with a first-order ellipticdifferential operator known as its Dirac operator.

If M is a Spinc manifold, then Td(M) is


If M is a complex-analyic manifold, then M has Chern classesc1, c2, . . . , cn and

expc1(M)/2A(M) = PTodd(c1, c2, . . . , cn)

EXAMPLE. Let M be a compact complex-analytic manifold.Set Ωp,q = C∞(M,Λp,qT ∗M)Ωp,q is the C vector space of all C∞ differential forms of type (p, q)Dolbeault complex

0 −→ Ω0,0 −→ Ω0,1 −→ Ω0,2 −→ · · · −→ Ω0,n −→ 0

The Dirac operator (of the underlying Spinc manifold) is theassembled Dolbeault complex

∂ + ∂∗ :⊕j

Ω0, 2j −→⊕j

Ω0, 2j+1


TWO POINTS OF VIEW ON SPINc MANIFOLDS

1. Spinc is a slight strengthening of oriented. The orientedmanifolds that occur in practice are Spinc.

2. Spinc is much weaker than complex-analytic. BUT theassempled Dolbeault complex survives (as the Dirac operator).AND the Todd class survives.

M Spinc =⇒ ∃ Td(M) ∈ H∗(M ;Q)

SPECIAL CASE OF ATIYAH-SINGERLet M be a compact even-dimensional Spinc manifoldwithout boundary. Let E be a C vector bundle on M .

DE denotes the Dirac operator of M tensored with E.

DE : C∞(M,S+ ⊗ E) −→ C∞(M,S− ⊗ E)

S+, S− are the positive (negative) spinor bundles on M .


K0(·)

Definition

Define an abelian group denoted K0(·) by considering pairs (M,E)such that:

1 M is a compact even-dimensional Spinc manifold withoutboundary.

2 E is a C vector bundle on M .

Set K0(·) = (M,E)/ ∼ where the the equivalence relation ∼is generated by the three elementary steps

Bordism

Direct sum - disjoint union

Vector bundle modification

Addition in K0(·) is disjoint union.

(M,E) + (M ′, E′) = (M tM ′, E t E′)

In K0(·) the additive inverse of (M,E) is (−M,E) where −Mdenotes M with the Spinc structure reversed.

−(M,E) = (−M,E)

Isomorphism (M,E) is isomorphic to (M ′, E′) iff ∃ adiffeomorphism

ψ : M →M ′

preserving the Spinc-structures on M,M ′ and with

ψ∗(E′) ∼= E.

Bordism (M0, E0) is bordant to (M1, E1) iff ∃ (Ω, E) such that:

1 Ω is a compact odd-dimensional Spinc manifold withboundary.

2 E is a C vector bundle on Ω.

3 (∂Ω, E|∂Ω) ∼= (M0, E0) t (−M1, E1, )

−M1 is M1 with the Spinc structure reversed.

(M0, E0) (−M1, E1)

Direct sum - disjoint union

Let E,E′ be two C vector bundles on M

(M,E) t (M,E′) ∼ (M,E ⊕ E′)

Vector bundle modification

(M,E)

Let F be a Spinc vector bundle on M

Assume thatdimR(Fp) ≡ 0 mod 2 p ∈M

for every fiber Fp of F

1R = M × R

S(F ⊕ 1R) := unit sphere bundle of F ⊕ 1R

(M,E) ∼ (S(F ⊕ 1R), β ⊗ π∗E)

S(F ⊕ 1R)

M

π

This is a fibration with even-dimensional spheres as fibers.

F ⊕ 1R is a Spinc vector bundle on M with odd-dimensional fibers.

The Spinc structure for F causes there to appear on S(F ⊕ 1R) aC-vector bundle β whose restriction to each fiber of π is the Bottgenerator vector bundle of that even-dimensional sphere.

(M,E) ∼ (S(F ⊕ 1R), β ⊗ π∗E)

Addition in K0(·) is disjoint union.

(M,E) + (M ′, E′) = (M tM ′, E t E′)

In K0(·) the additive inverse of (M,E) is (−M,E) where −Mdenotes M with the Spinc structure reversed.

−(M,E) = (−M,E)

DEFINITION. (M,E) bounds ⇐⇒ ∃ (Ω, E) with :

1 Ω is a compact odd-dimensional Spinc manifold withboundary.

2 E is a C vector bundle on Ω.

3 (∂Ω, E|∂Ω) ∼= (M,E)

REMARK. (M,E) = 0 in K0(·)⇐⇒ (M,E) ∼ (M ′, E′) where(M ′, E′) bounds.

Consider the homomorphism of abelian groups

K0(·) −→ Z(M,E) 7−→ Index(DE)

Notation

DE is the Dirac operator of M tensored with E.

It is a corollary of Bott periodicity that this homomorphism ofabelian groups is an isomorphism.

Equivalently, Index(DE) is a complete invariant for the equivalencerelation generated by the three elementary steps; i.e.(M,E) ∼ (M ′, E′) if and only if Index(DE) = Index(D′E′).

BOTT PERIODICITY

πjGL(n,C) =

Z j odd

0 j even

j = 0, 1, 2, . . . , 2n− 1

Why does Bott periodicity imply that

K0(·) −→ Z(M,E) 7−→ Index(DE)

is an isomorphism?

To prove surjectivity must find an (M,E) with Index(DE) = 1.

e.g. Let M = CPn, and let Ebe the trivial (complex) line bundle on CPnE=1C = CPn × CIndex(CPn, 1C) = 1

Thus Bott periodicity is not used in the proof of surjectivity.

Lemma used in the Proof of Injectivity

Given any (M,E) there exists an even-dimensional sphere S2n anda C-vector bundle F on S2n with (M,E) ∼ (S2n, F ).

Bott periodicity is not used in the proof of this lemma.The lemma is proved by a direct argument using the definition ofthe equivalence relation on the pairs (M,E).

Let r be a positive integer, and let VectC(S2n, r)be the set of isomorphism classes of C vector bundles on S2n

of rank r, i.e. of fiber dimension r.

VectC(S2n, r)←→ π2n−1GL(r,C)

PROOF OF INJECTIVITYLet (M,E) have Index(M,E) = 0.By the above lemma, we may assume that (M,E) = (S2n, F ).Using Bott periodicity plus the bijection

VectC(S2n, r)←→ π2n−1GL(r,C)

we may assume that F is of the form

F = θp ⊕ qβ

θp = S2n ×Cp and β is the Bott generator vector bundle on S2n.Convention. If q < 0, then qβ = |q|β∗.

Index(S2n, β) = 1 Index(S2n, θp) = 0Therefore

Index(S2n, F ) = 0 =⇒ q = 0

Hence (S2n, F ) = (S2n, θp). This bounds

(S2n, θp) = ∂(B2n+1, B2n+1 × Cp)

and so is zero in K0(·).QED

Define a homomorphism of abelian groups

K0(·) −→ Q(M,E) 7−→

(ch(E) ∪ Td(M)

)[M ]

where ch(E) is the Chern character of E and Td(M) is the Toddclass of M .

ch(E) ∈ H∗(M,Q) and Td(M) ∈ H∗(M,Q).

[M ] is the orientation cycle of M . [M ] ∈ H∗(M,Z).

Granted that

K0(·) −→ Z(M,E) 7−→ Index(DE)

is an isomorphism, to prove that these two homomorphisms areequal, it suffices to check one nonzero example.

Let X be a compact C∞ manifold without boundary.X is not required to be oriented.X is not required to be even dimensional.On X let

δ : C∞(X,E0) −→ C∞(X,E1)

be an elliptic differential (or pseudo-differential) operator.

(S(TX ⊕ 1R), Eσ) ∈ K0(·), and

Index(DEσ) = Index(δ).

(S(TX ⊕ 1R), Eσ)wwIndex(δ) = (ch(Eσ) ∪ Td((S(TX ⊕ 1R)))[(S(TX ⊕ 1R)]

and this is the general Atiyah-Singer formula.

S(TX ⊕ 1R) is the unit sphere bundle of TX ⊕ 1R.S(TX ⊕ 1R) is even dimensional and is — in a natural way — aSpinc manifold.

Eσ is the C vector bundle on S(TX ⊕ 1R) obtained by doing aclutching construction using the symbol σ of δ.

dirac operator - tata institute of fundamental...

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